106 
NATURE 
[Mov. 30, 1882 
ee 
the centre would be moved out through the face of the 
watch, and a blue pole in the opposite direction ; and the 
opposite would be the case if the current were reversed. 
This is easily remembered by those familiar with the re- 
presentation of couples in dynamics, by observing that 
when the direction of the current is the same as that in 
which a positive couple tends to turna body, the direction 
in which a red pole is urged is that in which the axis of 
the couple is drawn, Or, the direction of the force may 
be found at anv time, by remembering that the earth 
may be imagined to be a magnet turned into position by 
the action of a current flowing round the magnetic equator 
in the direction of the sun’s apparent motion. 
From the definition of a magnetic field we see that unit 
current may also be defined as that current which, flowing 
in a wire of unit length bent into an arc of a circle of unit 
radius, produces at the centre of the circle a magnetic 
field of unit intensity. The direction of the resultant 
magnetic force at that point is by Ampére’s law at right 
angles to the plane of the circle, and the side towards 
which it acts in any particular case may be found as 
stated above. 
If we take then the simple case of a single wire 
bent round into a circle and fixed in the magnetic 
meridian, with a magnet, whose dimensions are very 
small in comparison with the radius of the wire, hung 
by a torsionless fibre so as to rest horizontally with its 
centre at the centre of the circle, we may suppose that 
each pole of the magnet is at the same distance from all 
the elements of the wire. A current flowing in the wire 
acts, by Ampére’s theory, with a force on one pole of the 
needle towards one side of the plane of the circle, and 
on the other pole with an equal force toward the other side 
of that plane. The needle is thus acted on by a couple 
tending to turn it round, and it is deflected from its 
position of equilibrium until this couple is balanced by 
the return couple due to 7. Let us suppose the strength 
of each pole of the needle to be 7 units, 7 the radius of 
the circle, and C the strength of the current in it. Then 
by Ampére’s law we have for the whole force without 
regard to sign, exerted on either pole of the needle by the 
current, the value C7 77” or Cm *™, If 7 be the length of 
2 
the needle the couple is C7 /, before any deflection 
iP 
has taken place. After the needle has been deflected 
through the angle @ the arm / of the couple has become 
7 cos 6, and therefore the couple C m2 7 cos 6; and 
aa 
the return couple due to #7 is #H/ sin 6. Hence we 
have equilibrium when 
Cm7=1cos6=mH isin 6 
fo 
and therefore 
nr (8) 
27 
if @be the observed angle at which the needle rests in 
equilibrium when deflected as described trom the mag- 
netic meridian. If instead of a single circular turn of 
wire we had JV turns occupying an annular space of mean 
radius 7, and of dimensions of cross-section small com- 
pared with » we should have 
eee EE, (9) 
2a 
In practice the turns of wire of the tangent galvano- 
meter may not be all contained within such an annular 
space. It is necessary then to allow for the dimen- 
sions of the space occupied by the wire. For a coil 
made of wire of small section we may suppose that 
the actual current flowing across a unit of area is every- 
where the same. Hence if C be the current strength in 
each turn, and 7 the number of turns in unit area, we 
SEW Gg o Oo 6 
have for the current crossing the area 4 of an element & 
the value 27”C A. Taking a section of the coil 
through the centre, let &C be a radius drawn from 
the centre C in the plane cutting the coil into two 
equal and similar coils, and taking CD(= 2x) and 
DE(=y) at right angles to one another, we have 
A=dxdyand C E*= x?+y%. Hence the force exerted 
on a unit magnetic pole at the centre C by the ring sup- 
posed at right angles to the plane of the paper, of which 
this element is the section, will be 2 in the 
a 2 
direction at right angles to CZ and in the plane of the 
paper. If we call the component of this force at right 
angles to BC, dF, we have 
adFa27™ Cy dxdy 
+ 
Hence for the whole force at right angles to B C we have 
é rte 
/2 
Pazexc| [| none 
by bed (2? + y?)3 
- r—C. 
where 7 is the mean radius of the coil, 24 its breadth, 
and 2c its depth in the plane of the circle. 
Integrating, and putting V for the whole number of 
turns 47 6c, we get 
“ / j2 2 
FarNC lg or 
r—c+n(7 —- cf +6 
If 6 be the angle at which the deflecting couple is equili- 
brated by the return couple due to H, we have as before 
the equation 
. (10) 
va —eatanngs 
Hence, substituting the above value for / and solving for 
C, we have finally : 
cz {tan 6 
aN log rteot Vir + c) + 6° 
6 r—ct+ V7 — cf +e 
When the value of 7 is great in comparison with 6 and ¢ 
this reduces to the equation 
Caf riane (12) 
2aiNV 
which we found before by assuming all the turns to be 
contained in a small annular space of radius 7 In 
practice, in galvanometers used as standards for absolute 
ineasurements, generally neither 6 nor cis so great as 79 
of 7, and in these cases the difference between the values 
given by equations (11) and (12) is well within the limits 
of errors of observation, and the correction need not be 
made. The value of C given by (12) is then to be used. 
In this investigation the suspension fibre has been sup- 
posed torsionless. Ifa single fibre ot unspun silk is used 
as described below for this purpose, its torsion may for 
most practical purposes be sately neglected. The error 
produced by it may however be easily determined and 
atlowed for by turning the needle, supposed initially in 
the magnetic meridian, once or more times completely 
round, and noting its deviation from the magnetic meri- 
dian in its new position of equilibrium. The amount of 
this deviation, if any, may be easily observed by means 
of the attached index and divided circle, or reflected 
beam of light and scale, used as described below, to 
measure the deflections of the needle. From the result 
of this experiment the effect of torsion for any deflection 
may be calculated in the following manner. 
Let a be the angular deflection, in radian! measure, of 
the magnet from the magnetic meridian produced by 
turning the magnet once round, then the angle through 
which the thread has been twisted is 27— a, The couple 
produced by this torsion has for moment // /7 sin a. 
* A radian is the angle subtended at the centre of a circle by an arc equal 
in length to the radius. It has generally been called in bocks on trigono- 
metry hitherto by the ambiguous name wit angle in circular measure. 
(11) 
